InterMath/ Investigation/Geometry/Quadrilateral
Title: Interior Angles
Problem Statement:
What is the sum of the interior angles in a quadrilateral? Does the sum change if the
quadrilateral is convex? Does the sum change if the quadrilateral is concave? Make a conjecture
based on your observations and then prove your conjecture.
Problem setup:
We’re trying to determine if the sum of the interior angles in a quadrilateral. We’re also trying
to determine if the sum if the quadrilateral is concave.
Plans to Solve/Investigate the Problem:
Using the Geometer Sketchpad or paper, pencil, straight edge, and protractor, 4 groups of
students will construct a predetermine quadrilateral from the properties selected at random by the
students. After the construction of the quadrilateral, we will measure the interior angles to
determine the sum of the interior angles of several different quadrilaterals.
Investigation/Exploration of the Problem:
We have already seen that the sum of the angles in any triangle is 180º. So, we conclude that the
sum of the measures of the interior angles of any quadrilateral can be found by breaking any
quadrilateral into as many triangle as possible and then multiplying the number of triangle that
are created by the 180º. We can do this because the measure of the interior angles of any triangle
equals 180º, each of the two triangles will add 180º, to the total sum for the quadrilateral. So the
measure of the interior angles of a convex quadrilateral is the same as the sum of the measures of
the interior angles of two triangles, or 360º.
Slope AB = 0.00
m AB = 4.79 cm
A
B
m BC = 2.74 cm
m AD = 2.74 cm
Slope BC = -3.70
Slope AD = -3.70
parallelogram
D
m ABC = 105.11
C
m DC = 4.79 cm
m BCD = 74.89
m CDA = 105.11
Slope DC = 0.00
m DAB = 74.89
m ABC+m BCD+m CDA+m DAB = 360.00
First, I constructed a parallelogram; next I divided the parallelogram into triangles so that I could
measure the angles of the quadrilateral. Since this parallelogram has only one diagonal, it will be
divided into two triangles (180º + 180º =360º).
Slope EF = 0.00
F
E
trapezoid
G
H
Slope HG = 0.00
m FEH+m EHG+m HGF+m GFE = 360.00
m FEH = 120.39
m EHG = 59.61
m HGF = 65.22
m GFE = 114.78
A trapezoid has one diagonal. So it will be divided into two triangles (180º + 180º =360º).
m LM = 4.00 cm
N
O
Square
m NL = 4.00 cm
m OM = 4.00 cm
m NLM = 90.00
m LMO = 90.00
m MON = 90.00
m ONL = 90.00
L
m NL = 4.00 cm
M
m NLM+m LMO+m MON+m ONL = 360.00
A square has only one diagonal, so it will be divided into two triangles as well
(180º + 180º =360º).
A
E
m CAB = 80.89
m ABD = 122.57
C
m BDC = 37.20
B
Kite
H
m DCA = 119.34
F
Rhombus
m EFG = 101.80
m FGH = 75.07
m GHE = 105.22
m CAB+m ABD+m BDC+m DCA = 360.00
m HEF = 77.90
D
G
m EFG+m FGH+m GHE+m HEF = 360.00
The total measure of the interior angles of a
quadrilateral is 360 degrees.

Link to Sketch Pad demonstration of this conjecture
Remember that a polygon is convex if each of its interior angles is less than 180º. In other
words, the polygon is convex if it does not bend “inwards”.
The conjecture does not hold true for non-convex quadrilaterals.
Concave Quadrilateral
C
w
w = 4.10 cm
t = 5.57 cm
A
B
v = 5.36 cm
t
u
u = 7.79 cm
v
D
m  DAB = 16.61 
m  BCD = 32.65 
m  ABC = 131.58
m  CDA = 82.32 
m  DAB+m  ABC+m  BCD+m  CDA = 263.15
Extensions of the Problem:
Have students explore the sum of the measure of the interior angles of a polygon.
A
mJAB = 143.57
B
J
mABC = 143.81
mBCD = 144.50
mCDE = 142.11
mDEF = 144.53
mEFG = 146.89
I
C
mFGH = 141.34
mGHI = 146.06
mHIJ = 142.11
mIJA = 145.08
D
H
I multip li ed the number of tri angle s insi de the decagon
times 180 degrees
1808 = 1440.00
E
G
F
mJAB+mABC+mBCD+mCDE+mDEF+mEFG+mFGH+mGHI+mHIJ+mIJA = 1440.00
GPS Connection for Grades: 4-8
M4M2
Students will understand the concept of angle and how to measure angles.
a. Use tools, such as a protractor or angle ruler, and other methods, such as paper
folding or drawing a diagonal in a square, to measure angles.
b. Understand the meaning and measure of a half rotation (180°) and a full
rotation (360°).
M4G1
Students will define and identify the characteristics of geometric figures
through examination and construction.
c. Examine and classify quadrilateral (including parallelograms, squares,
rectangles, trapezoids, and rhombi)
d. Compare and contrast the relationships among quadrilaterals.
M7G1
Students will construct plane figures that meet given conditions.
b. Recognize that many constructions are based on the creation of congruent
triangles.
M4-8P1-5
Students will solve problems (using appropriate technology).
Students will reason and evaluate mathematical arguments.
Students will communicate mathematically.
Students will make connections among mathematical ideas and to other
disciplines
Students will represent mathematics in multiple ways.
Author & Contact
Norma Smith
norma_smith19@yahoo.com
Link(s) to resources, references, lesson plans, and/or other materials
http://www.geom.uiuc.edu/~dwiggins/conj06.html